In this example, we will solve the 1D heat equation

$\dfrac{\partial T}{\partial t} = \kappa \dfrac{\partial^2 T}{\partial x^2}$, with $\kappa = 1$

subject to the following boundary conditions:

$T(0,t)=0$ for $t \geq 0$
$\frac{\partial T}{\partial x}(x=l,t) = 0$ for $t \geq 0$
$T(x,0) = T_0 \sin\dfrac{3\pi x}{2l}$ for $0 \leq x \leq l$

To do this, we use the implicit Crank-Nicolson scheme.
This scheme is unconditionally stable, so we can choose any values of $\Delta t$ and $\Delta x$.

In the figures below, we vary $\Delta t$ and $\Delta x$ to see the effect on the solution.

Use the sliders vary $\Delta x$ and $\Delta t$ and compare the solutions here to those from the explicit time marching scheme in Lecture 5.

Implicit Crank-Nicolson Scheme

Fist, we change $\Delta t$ while keeping $\Delta x = 0.01$ fixed:

Next, we change $\Delta x$ while keeping $\Delta t = 5 \times 10^{-5}$ fixed: